Abstract
We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size c, the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone space KA of all such Boolean algebras A contains a copy of the Čech-Stone compactification of the integers βℕ and the Banach space C(KA has l∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.
Original language | English |
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Pages (from-to) | 305-312 |
Number of pages | 8 |
Journal | Algebra Universalis |
Volume | 69 |
Issue number | 4 |
DOIs | |
State | Published - Jun 2013 |
Keywords
- 46B10
- 54D
- Efimov's problem
- Grothendieck property
- independent families
- Primary: 06E
- Secondary: 46E15
- subsequential completeness property